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[sin a 1. In this question, you will be using the following trigonometric identities: cos? a + sin? a 1 (1) cos(a + 3) cos a cos 3 - sin a sin 3 (2) sin(a + B) = sin a cos 3 + cos a sin 3 (3) where a, 3 E R. You do not need to prove these identities. You may also use without proof the fact that the set { [cos a sin a :a € R is eractly the set of unit vectors in R2. Now for any real number a, define cos a - sin a Ra = sin a COS a (a) Prove that for all a, 3 E R, R R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R, Ra, for all a E R. (c) Prove that for all a ER and all x, y e R?, (R,x) · (Ray) =x•y (d) Suppose A is a 2 x 2 matrix such that for all x, y e R2, (Ax) (Ay) = x y Must it be true that A Ra, for some a E R? Either prove this, or give a counterexample (including justification). (e) Let B = be any 2 x 2 matrix. [cos a = u11 sin a (i) Show that there are real numbers uu and a such that Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a e R. Use the invertibility of Roa to prove that there are unique U12, U22 E R such that [cos a = u12 - sin a U22 COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a E R and some upper-triangular matrix U. (iv) Suppose that B RaU = R3V, where a, 3 ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.

[sin a 1. In this question, you will be using the following trigonometric identities: cos? a + sin? a 1 (1) cos(a + 3) cos a cos 3 - sin a sin 3 (2) sin(a + B) = sin a cos 3 + cos a sin 3 (3) where a, 3 E R. You do not need to prove these identities. You may also use without proof the fact that the set { [cos a sin a :a € R is eractly the set of unit vectors in R2. Now for any real number a, define cos a - sin a Ra = sin a COS a (a) Prove that for all a, 3 E R, R R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R, Ra, for all a E R. (c) Prove that for all a ER and all x, y e R?, (R,x) · (Ray) =x•y (d) Suppose A is a 2 x 2 matrix such that for all x, y e R2, (Ax) (Ay) = x y Must it be true that A Ra, for some a E R? Either prove this, or give a counterexample (including justification). (e) Let B = be any 2 x 2 matrix. [cos a = u11 sin a (i) Show that there are real numbers uu and a such that Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a e R. Use the invertibility of Roa to prove that there are unique U12, U22 E R such that [cos a = u12 - sin a U22 COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a E R and some upper-triangular matrix U. (iv) Suppose that B RaU = R3V, where a, 3 ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.3: Trigonometric Functions Of Real Numbers
Problem 43E
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1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) 1 cos(a + B) cos a cos 3 – sin a sin B %3D sin(a + B) = sin a cos 3 + cos a sin 3 (3) where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set [cos a sin a : αER is exactly the set of unit vectors in R2. Now for any real number a, define cos a - sin a sin a R. = COs a (a) Prove that for all a, 3 E R, R.R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R = R-a, for all a E R. (c) Prove that for all a ER and all x, y E R?, (R.x) · (Ray) =x y (d) Suppose A is a 2 x 2 matrix such that for all x, y € R?, (Ах) (Ау) 3D х у Must it be true that A = Ra, for some a e R? Either prove this, or give a counterexample (including justification). (e) Let B = be any 2 x 2 matrix. [cos a = U11 sin a (i) Show that there are real numbers u1u and a such that Нint: егpress as a scalar multiple of a unit vector, and hence find an expression for u1 in terms of a and c. (ii) Let a e R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that cos a = U12 sin a - sin a + u22 COs a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RU = R3V, where a, 3 ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) 1 cos(a + B) cos a cos 3 – sin a sin B %3D sin(a + B) = sin a cos 3 + cos a sin 3 (3) where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set [cos a sin a : αER is exactly the set of unit vectors in R2. Now for any real number a, define cos a - sin a sin a R. = COs a (a) Prove that for all a, 3 E R, R.R3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R = R-a, for all a E R. (c) Prove that for all a ER and all x, y E R?, (R.x) · (Ray) =x y (d) Suppose A is a 2 x 2 matrix such that for all x, y € R?, (Ах) (Ау) 3D х у Must it be true that A = Ra, for some a e R? Either prove this, or give a counterexample (including justification). (e) Let B = be any 2 x 2 matrix. [cos a = U11 sin a (i) Show that there are real numbers u1u and a such that Нint: егpress as a scalar multiple of a unit vector, and hence find an expression for u1 in terms of a and c. (ii) Let a e R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that cos a = U12 sin a - sin a + u22 COs a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RU = R3V, where a, 3 ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.
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